Properties

Label 2-8007-1.1-c1-0-363
Degree $2$
Conductor $8007$
Sign $-1$
Analytic cond. $63.9362$
Root an. cond. $7.99601$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.0787·2-s − 3-s − 1.99·4-s + 1.90·5-s − 0.0787·6-s + 2.89·7-s − 0.314·8-s + 9-s + 0.150·10-s + 5.10·11-s + 1.99·12-s − 0.339·13-s + 0.228·14-s − 1.90·15-s + 3.96·16-s + 17-s + 0.0787·18-s − 5.66·19-s − 3.80·20-s − 2.89·21-s + 0.401·22-s + 1.54·23-s + 0.314·24-s − 1.36·25-s − 0.0267·26-s − 27-s − 5.77·28-s + ⋯
L(s)  = 1  + 0.0556·2-s − 0.577·3-s − 0.996·4-s + 0.852·5-s − 0.0321·6-s + 1.09·7-s − 0.111·8-s + 0.333·9-s + 0.0475·10-s + 1.53·11-s + 0.575·12-s − 0.0942·13-s + 0.0610·14-s − 0.492·15-s + 0.990·16-s + 0.242·17-s + 0.0185·18-s − 1.29·19-s − 0.850·20-s − 0.632·21-s + 0.0856·22-s + 0.322·23-s + 0.0642·24-s − 0.272·25-s − 0.00525·26-s − 0.192·27-s − 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8007 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8007\)    =    \(3 \cdot 17 \cdot 157\)
Sign: $-1$
Analytic conductor: \(63.9362\)
Root analytic conductor: \(7.99601\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8007,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
17 \( 1 - T \)
157 \( 1 - T \)
good2 \( 1 - 0.0787T + 2T^{2} \)
5 \( 1 - 1.90T + 5T^{2} \)
7 \( 1 - 2.89T + 7T^{2} \)
11 \( 1 - 5.10T + 11T^{2} \)
13 \( 1 + 0.339T + 13T^{2} \)
19 \( 1 + 5.66T + 19T^{2} \)
23 \( 1 - 1.54T + 23T^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 + 0.163T + 31T^{2} \)
37 \( 1 - 2.75T + 37T^{2} \)
41 \( 1 + 2.57T + 41T^{2} \)
43 \( 1 + 12.3T + 43T^{2} \)
47 \( 1 + 2.03T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 14.0T + 59T^{2} \)
61 \( 1 - 2.12T + 61T^{2} \)
67 \( 1 + 2.69T + 67T^{2} \)
71 \( 1 + 15.7T + 71T^{2} \)
73 \( 1 + 7.30T + 73T^{2} \)
79 \( 1 + 1.02T + 79T^{2} \)
83 \( 1 - 6.13T + 83T^{2} \)
89 \( 1 - 9.63T + 89T^{2} \)
97 \( 1 - 5.26T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.56754914992164575868040924256, −6.52989760737588754116341055996, −6.08953504700389065004839774144, −5.33534800364368974755805152597, −4.67577024813270797166463152046, −4.17994137886787588776816249690, −3.29624779684687788266880217179, −1.74124592073216046155539166504, −1.47385603427780621703285722989, 0, 1.47385603427780621703285722989, 1.74124592073216046155539166504, 3.29624779684687788266880217179, 4.17994137886787588776816249690, 4.67577024813270797166463152046, 5.33534800364368974755805152597, 6.08953504700389065004839774144, 6.52989760737588754116341055996, 7.56754914992164575868040924256

Graph of the $Z$-function along the critical line